BALTIC-WAY 1998 Problem 8

Let $P_{k}(x)=1+x+x^{2}+\cdots+x^{k-1}$. Show that

$$\sum_{k=1}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) P_{k}(x)=2^{n-1} P_{n}\left(\frac{1+x}{2}\right)$$

for every real number $x$ and every positive integer $n$.